where \(\tilde{N}(\cdot ,\cdot )\)  is a Poisson random martingale measure. Under standard assumptions, Tang and Li [1]} first proved their existence and uniqueness results for general BSDEPs, which are derived as adjoint equations when they studied the global maximum principle for the controlled stochastic differential equation with random jumps (SDEP for short). Wu [2]} studied the existence and uniqueness result of the solutions to fully coupled forward-backward SDEP (FBSDEP for short). Situ [3]} first obtained the global maximum principle for the controlled SDEP where the control variable is not included in the jump coefficient. Tang and Li [1]} completely proved the global maximum principle where the control variable enters into both diffusion and jump coefficients. Shi and Wu [5]} extended the result of Peng [6]} to discontinuous case, and proved a local maximum principle for a kind of controlled FBSDEPs. Recently, Song et al. [7]} found that, in the work of [1]}, the third estimate in (2.10) is not correct, which was also put forward by Situ [9]} in the early, and the subsequent variational calculus then underestimated the influence of jumps. In [7]}, they overcome the deficiencies with a new spike variation technique and obtained a new global maximum principle for controlled SDEPs.
